Question

К окружности проведены касательная и секущая из одной точки m. Касательная касается окружности в точке N, секущая пересекает окружность в точках P и Q. Известно что MP=4, PQ=5 Найдите MN

Answer 1

MN^2=MQ×MP=9×4=336 MN=корень36 MN=6 Ответ: 6

Answer 2

Problem Analysis

We are given a circle with a tangent and a secant line passing through a point M. The tangent line touches the circle at point N, and the secant line intersects the circle at points P and Q. We are also given that MP = 4 and PQ = 5. We need to find MN.

Solution

To find MN, we can use the property that the length of a tangent segment from an external point to a circle is equal to the length of the secant segment squared. In other words, MN^2 = MP * MQ.

Given that MP = 4 and PQ = 5, we can find MQ using the equation MQ = MP + PQ.

Let's calculate MQ and then find MN.

Calculation

1. MQ = MP + PQ = 4 + 5 = 9. 2. MN^2 = MP * MQ = 4 * 9 = 36. 3. Taking the square root of both sides, we get MN = √36 = 6.

Answer

Therefore, MN = 6.

Note: The calculation assumes that the lengths of the tangent and secant segments are positive.